Mathematical Theory and Modeling ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.2, No.3, 2012
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On the Solution of Incompressible Fluid Flow Equations: a Comparative Study Kumar Dookhitram1* Roddy Lollchund2 1.
Department of Applied Mathematical Sciences, School of Innovative Technologies and Engineering, University of Technology, Mauritius, La Tour Koenig, Pointe-aux-Sables, Republic of Mauritius
2.
Department of Physics, Faculty of Science, University of Mauritius, Réduit, Republic of Mauritius
* E-mail of the corresponding author:
[email protected] Abstract This paper studies and contrasts the performances of three iterative methods for computing the solution of large sparse linear systems arising in the numerical computations of incompressible Navier-Stokes equations. The emphasis is on the traditional Gauss-Seidel (GS) and Point Successive Over-relaxation (PSOR) algorithms as well as Krylov projection techniques such as Generalized Minimal Residual (GMRES). The performances of these three solvers for the second-order finite difference algebraic equations are comparatively studied by their application to solve a benchmark problem in Computational Fluid Dynamics (CFD). It is found that as the mesh size increases, GMRES gives the fastest convergence rate in terms of cpu time and number of iterations. Keywords: viscous flows, Navier-Stokes equations, linear system, iterative, GMRES 1. Introduction Consider the system of viscous incompressible flows governed by the unsteady Navier-Stokes equations, which are given (in non-dimensional form) as ,
(1)
(2) Our notation is standard: u is the fluid velocity; p is the pressure; Re is the Reynolds number and f represents a body force. The fluid is assumed to fill a two-dimensional square domain with boundary , where subscript D stands for Dirichlet. The flow is specified by the no-slip condition on , where is the velocity on the boundary. In this paper we study the well known Stokes lid-driven cavity problem, i.e., the flow of an incompressible fluid within an enclosed square cavity driven by a sliding lid at constant speed (see Fig. 1). In our experiment the vertical velocity v is set to zero everywhere, whereas the horizontal velocity component u is set to unity on the lid, and is zero on the other boundaries. One interesting feature of this flow is that the pressure is singular at the top corners, i.e., where the imposed velocity is discontinuous. This flow problem has long served as a benchmark in the validation of two-dimensional numerical solution methods for incompressible flows (Ferziger & Peric 2002). It is of great interest in the CFD community since it mimics many aeronautical, environmental and industrial flows such the flow over structures in airfoils or the cooling flow over electronic devices (Povitsky 2001). 2. Procedure
We discretize
with uniform and equal mesh sizes of dimension h and denote 9
as
Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.2, No.3, 2012 the number of grid points along x and y coordinates. Moreover, we use the index pair (i, j) to represent the mesh point (xi, yj), with and for . On such a uniform mesh, the numerical solution of (1)-(2) with second-order central-difference discretization introduces an explicit discretised Poisson-type equation for the pressure field with valid boundary conditions ( ) as follows: ,
(3)
where and are intermediate velocity components obtained by a Hodge Decomposition of (1)-(2), is the time step and the superscript is the time level (Moreno & Ramaswamy 1997). Such type of Pressure-Poisson equation (PPE) is obtained when a projection scheme such as SIMPLE, PISO or Fractional Step Method is employed. In literature, it has been reported that in finding a numerical solution for (1)-(2), the major computational cost is spent in the calculation part which involves (3) (Patankar 1980; Johnston & Liu 2004,). In the sequel, our aim is to compare the performances of some iterative methods to solve (3). Following the conventional process, (3) can be written as the linear system (4) is block tri-diagonal (Fig. 2). The elements of x consist of the pressure at where the coefficient matrix each point on the grid and b is obtained explicitly from the right hand side of (3). 3. Experiments We compare the performances of three iterative methods to solve the linear system (4). The linear solvers considered here are the Gauss-Seidel, Point Successive Over Relaxation and the Krylov projection Generalized Minimal Residual (Saad & Schultz 1986) methods. The objective is to study their convergence rate and cpu time. The convergence criteria for determining steady state is based on the difference between two successive iterates; whereby the procedure for finding the fluid velocity u is stopped when this difference, measured by L1 norm is less than a prescribed tolerance (tol). The numerical experiments are performed for uniform grids with , , , and points for . Tables (1)-(3) show the number of iterations required (k) and the computational time (cpu) in seconds for each solver to reach steady state. It can be observed from these tables that the GMRES method is more efficient than the GS and PSOR methods. For low Reynolds number (100 and 400), when the mesh size is above , the PSOR method is about 1.2 times faster than the GS method, while the GMRES method is more than 2.5 times faster than the PSOR method. For higher Reynolds number (1000), the GS and PSOR methods are comparable in terms of computational time required to reach steady state. However, GMRES gains over GS and PSOR by more than 1.3 times as from the grids. It is interesting to note that for all cases studied, the speed-up of GMRES for the fine mesh is much more efficient than for the coarse mesh. Fig. 3 displays the velocity vector and pressure fields of the lid-driven cavity flow at calculated on a grid. It can be observed that the moving lid creates a strong vortex and a sequence of weaker vortices in the lower two corners of the cavity. 4. Conclusions In this paper, we demonstrated that for the numerical solution of incompressible flows, the use of GMRES for solving the linear PPE is much more efficient than the traditional GS and PSOR schemes. In practice, one always uses a fine mesh to obtain a better accuracy of the required solution. Following the conventional process, we observed that for large grid size GMRES converges faster than other methods. Moreover, the use of GMRES provides very accurate numerical results by using very little cpu time and virtual storage.
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Mathematical Theory and Modeling ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.2, No.3, 2012
www.iiste.org
References Ferziger J. H. & Peric M. (2002), Computational methods for fluid dynamics, 3rd edn, Springer. Johnston H. & Liu J. (2004), “Accurate, stable and efficient Navier-Stokes solvers based on explicit treatmentof the pressure term”, Journal of Computational Physics 199, 221–259. Moreno R. & Ramaswamy B. (1997), “Numerical study of three-dimensional incompressible thermal flows in complex geometries. Part I: theory and benchmark solutions”, International Journal for Numerical Methods for Heat and Fluid Flow 7(4), 297–343. Patankar S. V. (1980), Numerical heat transfer and fluid flow, Taylor and Francis. Povitsky A. (2001), “Three-dimensional flow in cavity at Yaw”, ICASE Report 2001-31. Saad Y. & Schultz M. H. (1986), “GMRES: A generalised minimal residual algorithm for solving nonsymmetric linear systems”, SIAM J. Sci. Stat. Comput. 7, 856–869.
Figure 1. Geometry of the lid-driven cavity problem in the domain
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Mathematical Theory and Modeling ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.2, No.3, 2012
Figure 2. The coefficient matrix
Figure 3. Flow and pressure fields at
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of the linear system (4)
calculated on a
12
grid
Mathematical Theory and Modeling ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.2, No.3, 2012
www.iiste.org
Table 1. Performance of three iterative methods for PPE equation (Re = 100, tol = 10-6) Grids
GS
PSOR
GMRES
k
cpu
k
cpu
k
cpu
310
5.5
317
5.0
310
9.5
433
20.4
444
17.7
434
19.4
562
57.5
574
49.6
562
38.4
1023
420.3
1032
362.7
1025
173.1
1862
1946.7
1869
1687.6
1863
557.6
Table 2. Performance of three iterative methods for PPE equation (Re = 400, tol = 10-6) Grids
GS
PSOR
GMRES
k
cpu
k
cpu
k
Cpu
733
8.6
727
7.7
723
21.5
1011
31.9
1004
28.1
1003
42.2
1305
91.6
1300
81.6
1305
80.2
2065
617.9
2048
545.5
2046
306.3
2794
2375.5
2788
2117.8
2790
771.4
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Mathematical Theory and Modeling ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.2, No.3, 2012
www.iiste.org
Table 3. Performance of three iterative methods for PPE equation (Re = 1000, tol = 10-6) Grids
GS
PSOR
GMRES
k
cpu
k
cpu
k
Cpu
1249
13.0
1270
11.9
1247
37.5
1712
44.4
1705
40.8
1705
72.2
2245
118.0
2280
110.9
2231
133.0
3701
721.0
3640
679.9
3643
503.1
5327
2819.9
5231
2748.5
5225
1323.6
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